Tipping elements, such as the Greenland Ice Sheet, the Atlantic meridional ocean circulation (AMOC) or the Amazon rainforest, interact with one another and with other non-linear systems such as the El-Nino Southern Oscillation (ENSO). In doing so the risk of any one element collapsing into a degraded state can be drastically affected, typically increasing due to the interactions. In this work, therefore, we propose a fully probabilistic network model for risk assessment of interacting tipping elements that coherently incorporates literature-based belief assessments of intra-element interactions. We provide analytic results for the equilibrium risks of nine interacting tipping elements, the existence and stability of their stationary distributions and convergence times to the equilibrium solution. Moreover we simulate their tipping risks until 2350 using emission pathways from the shared socio-economic pathways (SSP 1-1.9, 1-2.6, 2-4.5, 3-7.0, and 5-8.5). Compared to the hypothetical no-interactions case, we find that interactions tend to destabilise the climate system, for instance the coral reefs are likely to have collapsed by 2100 even under the most optimistic scenario (SSP1-1.9). The effects of interactions, however, are most noticeable after 2100, especially for the highest shared socio-economic pathways (SSP3-7.0 and SSP5-8.5). In summary, our comprehensive risk assessment framework for tipping elements indicates that rapid mitigation is essential to keep temperatures as close as possible to 1.5 °C in the short term and below 1 °C in the longer run.